Identification of Semimartingale Neuronal Systems

In this paper a nerve cell (or neuron) is modeled as a system that transforms a semimartingale input into a point process output. This extends a result of van der Boogaard et al. (1986) in which neurons transform point process inputs into point process outputs. The semimartingale input model proposed here encompasses Poisson, Gauss-Poisson, diffusion and Ito-Markov processes which have been considered in the literature. Using Ito's formula, the intensity (or pulse generation function) of the output point process is expressed explicitly in terms of the input process under mild differentiability conditions on the intensity. The conditional mean and quadratic variation of the output process are derived in terms of the corresponding mean and quadratic variation of the input process. Some related problems of parameter estimation are also considered.